Dynamical Systems. Proceedings of a Symposium Held at the by M. M. Peixoto

By M. M. Peixoto

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Extra info for Dynamical Systems. Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26–august 14, 1971

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For any y e V G, sufficiently close to x, TyM splits as TyM = Ey* Q)Eyx\ where dtpgo contracts vectors of Ey* and expands those of E*. Define E* = \JyeVG, £ / , Eu = \Jy€yG, E*. Since dxpq sends E* into E\y and commutes with άψ^ the splitting E8 0 TV 0 Eu is continuous and d

Nl) = e x p f o ^ + · · · + ί*λΛ)μΐι · · · /ip. If k > 1, ρ is structurally stable if and only if some X-t φ 0. This means that ρ is transitive on each component of E — {0}. If k = 0, ρ is not transitive. On the other hand, we notice that if log μ,, . . , log μι are linearly independent over the rationals, the orbit by ρ of some v e E — {0} is dense in an open subset of E. If they are linearly dependent over the rationals, &ν(ρ) becomes discrete in E — {0}. Therefore, ρ is not stable. Suppose dim E = 2.

Then for any x e VG,, there is a neighborhood Aof x such that {i^ Π A} is an invariant C 1 fibration ofWnA over F ö / n A. 1 there is a neighborhood ^4 of x such that {2^ n A} is a C1 fibration ofWnA over F ö / n ^4 invariant by 9? 10 and the next two propositions. 3 Let x e M be a hyperbolic fixed point of an action φ : G X M -► M, rank G > dim M. Let ρ : G - * Aut(7yif ) be the linear action induced by φ at x. Then there is a neighborhood [/ of x and a homeomorphism / / : U —► T^M such that H οφτ = ρτ o H.

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